aa r X i v : . [ m a t h . C V ] F e b ON A PROBLEM OF CHIRKA
SŁAWOMIR DINEW AND ŻYWOMIR DINEW
Abstract.
We observe that a slight adjustment of a method of Caffarelli, Li,and Nirenberg yields that plurisubharmonic functions extend across subhar-monic singularities as long as the singularities form a closed set of measurezero. This solves a problem posed by Chirka.
Introduction
Let Ω be a domain in C n and E ⊆ Ω be a closed subset of Lebesgue measurezero. Let u be a subharmonic function in Ω which is furthermore plurisubharmonicin Ω \ E . Then u is said to be a plurisubharmonic function with subharmonicsingularities along E . It is natural to ask under what conditions on E will u bea genuine plurisubharmonic function on the whole Ω . Alternatively, one can askwhen does the positive (1 , -current i∂∂u on Ω \ E extend past E provided itstrace measure i∂∂u ∧ β n − , where β := i∂∂ k z k , extends past E as a positiveBorel measure. Such questions appear in potential theory on Teichm¨uller spaces-[11] and in extension problems of positive line bundles past small sets- [14, 8].The problem is interesting only for sufficiently large sets E , as plurisubharmonicfunctions automatically extend past closed sets of (2 n − -dimensional Hausdorffmeasure zero- [8].In 2003, Chirka proved in [5] the following result: Theorem 0.1.
Let E be a real hypersurface of class C in Ω . Then any function u of the above type extends past E . Under additional smoothness assumptions this was previously shown in [1], seealso [12] for a survey on these matters.Note that the subharmonicity on Ω is essential here as the example u ( z ) := ( k z k if k z k ≤ if k z k > shows. Unfortunately, as mentioned in [5], the method applied there cruciallydepends on the C regularity. Thus in [5] the following conjecture was made: Any plurisubharmonic function with subharmonic singularities along E extendsas a plurisubharmonic function provided E is a closed subset of Ω with a locallyfinite (2 n − -dimensional Hausdorff measure. Mathematics Subject Classification.
Primary 32U30; Secondary 32U40, 31B05, 32D20,32U15, 35D40.
Key words and phrases. plurisubharmonic function, removable set, viscosity theory.The first Author was supported by the Polish National Science Centre grant2017/26/E/ST1/00955.
The aim of this note is to solve affirmatively this problem. In fact we observethat a slight modification of the tools introduced by Caffarelli, Li, and Nirenbergyields a much stronger result:
Theorem 0.2 (Main Result) . Let E ⊆ Ω be a closed subset of Lebesgue measurezero. Then any subharmonic function u in Ω which is plurisubharmonic in Ω \ E is actually plurisubharmonic in the whole Ω . Proof of the main result
Of course there is nothing to prove in complex dimension .We begin with a classical result (see [10], Proposition 3.2.10’) that subharmonic-ity and plurisubharmonicity can be tested through C -smooth local majorants. Lemma 1.1.
Let Ω ⊆ C n be a domain. An upper semicontinuous function u on Ω is subharmonic (respectively plurisubharmonic) if for every z ∈ Ω and every local C -smooth function ϕ defined near z and satisfying ϕ ( z ) ≥ u ( z ) with equality at z we have ∆ ϕ ( z ) ≥ (respectively we have ∂ ϕ∂z j ∂ ¯ z k ( z ) ≥ ). Note that ϕ above is subject to conditions provided that it exists. In particular,the assumption can be void at some points z ∈ Ω . Also note that in the languageof viscosity theory the above lemma says that u is subharmonic iff it is a viscositysubsolution to the Laplace equation ∆ v = 0 (that is ∆ u ≥ ) and u is plurisubhar-monic iff it is a viscosity subsolution to the constrained complex Hessian equation det + (cid:16) ∂ v∂z j ∂ ¯ z k (cid:17) = 0 (that is det + (cid:16) ∂ u∂z j ∂ ¯ z k (cid:17) ≥ ), where det + ( A ) = ( det( A ) if A ≥ −∞ otherwise . For more details regarding the viscosity theory of such constrained complex Hessianequations we refer to [6].The proof of the main result is essentially contained in Theorem 1.2 in [4] oncewe adjust the tools applied there to the constrained complex equation det + (cid:18) ∂ u∂z j ∂ ¯ z k (cid:19) = 0 . We provide the details for the sake of completeness. We learned about this argumentthrough [2] and [15].As a direct corollary of Lemma 1.1 plurisubharmonicity of u would follow if onecan show that for any z ∈ E and any local C majorant ϕ ≥ u , ϕ ( z ) = u ( z ) onehas ∂ ϕ∂z j ∂ ¯ z k ( z ) ≥ as matrices.Translating if necessary one may assume that z is the origin, that Ω contains aball B δ centered at the origin and that ϕ is defined on B δ . For a fixed < δ < δ we consider the function v δ ( z ) := ϕ ( z ) + δ k z k − δ − u ( z ) . N A PROBLEM OF CHIRKA 3
By the very definition of ϕ we have v δ ( z ) ≥ on the collar B δ \ B δ . Also, as u issubharmonic, v δ is bounded below on B δ and v δ is a lower semicontinuous viscositysupersolution to the Poisson equation ∆ v = ∆ ϕ + 4 nδ =: f , that is ∆ v δ ≤ f. Note that f is continuous.Consider the following convex envelope in B δ Γ v δ ( z ) := sup { l ( z ) | l − affine , l ≤ v δ on B δ , l ≤ on B δ \ B δ } . As v δ is bounded below the family of functions l above is non void.The following Alexandrov-Bakelman-Pucci type theorem for viscosity superso-lutions (see Theorem 3.2 in [3]) is crucial: Lemma 1.2.
Let v δ and Γ v δ be as above. Then for some universal constant C ,dependent only on n we have δ ≤ sup B δ | v δ | ≤ Cδ Z { B δ ∩{ v δ =Γ vδ }} max { f, } n dV ! n . We remark that this result is stated in [3] for continuous supersolutions but theproof there works for lower semicontinuous supersolutions as well. Here we cruciallyexploit the fact that the Laplacian is a uniformly elliptic operator.The upshot is that for every δ ∈ (cid:0) , δ (cid:1) the function v δ matches its convex enve-lope Γ v δ on a set of positive measure within B δ . As E is of measure zero we picka point z δ ∈ { B δ ∩ { v δ = Γ v δ }} \ E . As u is plurisubharmonic around z δ for any < r small enough and any unit vector T ∈ C n we have u ( z δ ) ≤ π Z π u ( z δ + re iθ T ) dθ. The same inequality is valid for the convex (hence plurisubharmonic) function Γ v δ .But then ϕ ( z δ ) = u ( z δ ) + δ − δ k z δ k + Γ v δ ( z δ ) ≤ δ − δ k z δ k + 12 π Z π [ u + Γ v δ ]( z δ + re iθ T ) dθ. ≤ δ − δ k z δ k + 12 π Z π (cid:2) ϕ ( z δ + re iθ T ) − δ + δ k z δ + re iθ T k (cid:3) dθ = 12 π Z π ϕ ( z δ + re iθ T ) dθ + δr . After dividing by r and then letting r ց + we obtain n X j,k =1 ∂ ϕ∂z j ∂ ¯ z k ( z δ ) T j ¯ T k ≥ − δ. Finally, as δ ց + we have z δ → z ) and because ϕ is C it holds n X j,k =1 ∂ ϕ∂z j ∂ ¯ z k ( z ) T j ¯ T k ≥ . As T is an arbitrary unit vector this shows the non negative definiteness of thecomplex Hessian of ϕ at z . SŁAWOMIR DINEW AND ŻYWOMIR DINEW
Remark . Similar ideas are utilized in [9] to prove extension theorems but withthe assumption of boundedness or continuity instead of subharmonicity.
Remark . It is well known that there exist closed sets of Lebesgue measurezero and of full Hausdorff dimension. Take for example a product of n copiesof A , where A = { } ∪ S j ≥ A j and A j is a generalized Cantor set of Hausdorffdimension − /j situated in the interval [1 − /j, − / ( j + 1)] . We see thereforethat, unlike in many similar removable singularity theorems, there is no Hausdorffdimension threshold up to which our theorem holds. Also often the possibility toextend objects is related to the vanishing of some capacity. Closed sets of measurezero can have positive capacity, as again the product of Cantor sets demonstrates.This is another distinctive feature of our theorem. Remark . Our theorem could be combined with some removable singularitytheorems for (particular classes of) subharmonic functions, see e.g. [7],[13]. Now if u is subharmonic (of a particular class) on Ω \ F , where F is removable (for thisparticular class) and plurisubharmonic on Ω \ ( E ∪ F ) , where E is closed and ofmeasure zero, then u is plurisubharmonic on Ω . Remark . We note that the closedness assumption on E is not really necessary,and is introduced only to ensure that plurisubharmonicity on Ω \ E makes sense.Alternatively, we could assume that E is any set of Lebesgue measure zero and if u is plurisubharmonic on some neighborhood of Ω \ E then u is plurisubharmonicon Ω if it is subharmonic there. This may be essential in some situations, as theclosure of a null set can have positive measure. References
1. P. Blanchet,
On removable singularities of subharmonic and plurisubharmonic functions ,Complex Variables (1995), 311–322.2. J. E. M. Braga, D. Moreira, Zero Lebesgue measure sets as removable sets for degenerate fullynonlinear elliptic PDEs , NoDEA Nonlinear Differ. Equ. Appl. (2018). no. 2, Paper No.11, 12 pp.3. L. Caffarelli, X. Cabr´e, Fully Nonlinear Elliptic Equations , American Mathematical SocietyColloquium Publications, American Mathematical Society, Providence, RI, (1995).4. L. Caffarelli, Y. Li and L. Nirenberg,
Some remarks on singular solutions of nonlinear ellipticequations III: viscosity solutions including parabolic operators , Comm. Pure Appl. Math. (2013), no. 1, 109–143.5. E. M. Chirka, On the removal of subharmonic singularities of plurisubharmonic functions ,Ann. Polon. Math. (2003), 113–116.6. P. Eyssidieux, V. Guedj and A. Zeriahi, Viscosity solutions to degenerate complex Monge-Amp`ere equations , Comm. Pure Appl. Math. (2011), no. 8, 1059–1094.7. S. Gardiner, Removable singularities for subharmonic functions , Pacific J. Math. (1991),no. 1, 71–80.8. R. Harvey,
Removable singularities for positive currents , Amer. J. Math. (1974), 67–78.9. R. Harvey, B. Lawson, Removable singularities for nonlinear subequations , Indiana Univ.Math. J. (2014), no. 5, 1525–1552.10. L. H¨ormander, Notions of Convexity , Progress in Mathematics, . Birkh¨auser Boston, Inc.,Boston, MA, (1994).11. H. Miyachi,
Pluripotential theory on Teichm¨uller space I: Pluricomplex Green function , Con-form. Geom. Dyn. (2019), 221–250.12. J. Riihentaus, Removability results for subharmonic functions, for harmonic functions andfor holomorphic functions , Mat. Stud. (2016), no. 2, 152–158.13. A. Sadullaev, Zh. Yarmetov, Removable singularities of subharmonic functions in the classLip α . (Russian. Russian summary) Mat. Sb. (1995), no. 1, 131–148; translation in Sb.Math. (1995), no. 1, 133–150. N A PROBLEM OF CHIRKA 5
14. B. Shiffman,
Extension of positive line bundles and meromorphic maps , Invent. Math. (1972), no. 4, 332–347.15. A. Święch, A note on the upper perturbation property and removable sets for fully nonlineardegenerate elliptic PDE , NoDEA Nonlinear Differ. Equ. Appl. (2019), no. 1, Paper No. 3,4 pp. Faculty of Mathematics and Computer Science, Jagiellonian University 30-348 Krakow,Łojasiewicza 6, Poland
Email address : [email protected] Faculty of Mathematics and Computer Science, Jagiellonian University 30-348 Krakow,Łojasiewicza 6, Poland
Email address ::